To arrange the expressions by growth rate from slowest to fastest, we need to compare the rates at which these expressions increase as the input size, denoted by 'n', gets larger.
Let's analyze each expression one by one:
1. 10n: This expression grows exponentially with a base of 10. As 'n' increases, the value of 10n increases rapidly. However, it grows slower than 20n or 2n², as the base is smaller. Therefore, it is slower than both of them.
2. 5nlogn: This expression has a growth rate that lies between the linear (n) and quadratic (n²) growth rates. The logarithm term, logn, grows slower than any power of 'n', but it still contributes to the overall growth rate. So, it is slower than 10n but faster than n.
3. n!: This expression represents the factorial of 'n', which is the product of all positive integers from 1 to 'n'. The factorial function grows very rapidly as 'n' increases. It grows faster than any polynomial function like n or n². Hence, it is faster than both of them.
4. 20n: This expression grows linearly with a coefficient of 20. It increases at a constant rate as 'n' increases. It is faster than 10n but slower than 2n².
5. 2n²: This expression grows exponentially with a base of 2, and it also has a quadratic term. As 'n' increases, the value of 2n² increases at a much faster rate compared to any of the previous expressions. It is faster than all the expressions mentioned above.
So, the expressions arranged by growth rate from slowest to fastest are: 10n, 5nlogn, n!, 20n, 2n².